# Dependence of Hall coefficient on temperature in semiconductors

• Roger Dalton
In summary, the conversation discusses the temperature dependence of the Hall coefficient and how it can be calculated using the expressions for electron and hole densities. The ionisation energies $E_d$ and $E_a$ and the energy gap of the semiconductor $E_g$ can also be calculated from the final expression of the Hall coefficient. It is also mentioned that the donor and acceptor densities can be found from the Hall coefficient expression.
Roger Dalton
Homework Statement
By measuring the electrical conductivity and the Hall effect as a function of temperature, many characteristic parameters of semiconductors can be determined. Since charge transport in semiconductors takes place through both electrons in the conduction band and holes in the valence band, the two-band model expression for the Hall coefficient must be used.

(a) By measuring the temperature dependence of the Hall coefficient, how can the gap Eg of an n-type semiconductor be determined, as well as the distance Ed from the donor level to the edge of the conducting band? And for a p-type semiconductor, how can the distance Ea from the acceptor level to the valence band edge be determined?

(b) Can the nD density of the donors in an n-type semiconductor or the nA density of the acceptors in a p-type semiconductor be determined by measuring the Hall effect? If so, over what temperature range should the measurement be made?
Relevant Equations
The Hall coefficient for a semiconductor, in terms of the mobilities and the densities of the charge charriers (electrons and holes), is given by:

$$R_H = \frac{1}{e} \frac{p_v\mu_h^2-n_c\mu_e^2}{(p_v\mu_h+n_c\mu_e)^2}$$

Where $\mu_h$ and $\mu_e$ are the mobilities for holes and electrons respectively, and $p_v$ and $n_c$, their respective densities as well.
My first assumption is that the temperature dependence on the mobilities can be neglected, and so we would have:

$$R_H(T)= \frac{1}{e} \frac{p_v(T)\mu_h^2-n_c(T)\mu_e^2}{(p_v(T)\mu_h+n_c(T)\mu_e)^2}$$

The expression for the electron and hole densities could be derived from

$$\frac{n_c(n_c+n_A)}{n_D-n_A-n_c}=n_c^{eff}(T)e^{-E_d/k_BT} (1)$$
$$\frac{p_v(p_v+p_A)}{p_D-p_A-p_v}=p_v^{eff}(T)e^{-E_a/k_BT} (2)$$

Where $E_d$ is the ionisation energy it takes to extract an electron from the donor state to put it into the conduction band, $E_a$ is the ionisation energy it costs to extract a hole to put it in the valence band, and $n_D$, $p_D$, $n_A$, and $p_A$ are the densities of the donor and acceptor densities depending if they are electrons or holes. In this case, we can neglect both $n_A$ and $p_D$.

$$n_c^{eff}(T)=2\left(\frac{m_e^*k_BT}{2\pi\hbar^2}\right)^{3/2}$$
$$p_v^{eff}(T)=2\left(\frac{m_h^*k_BT}{2\pi\hbar^2}\right)^{3/2}$$

Solving (1) and (2) in terms of $n_c(T)$ and $p_v(T)$ and pluging the final expressions into the one for the Hall coeffcient would give us the dependence of Hall coefficient on temperature. Is this approach correct? The thing is that using this approach I do not know what the donor and acceptor densities are.

Besides, how could I calculate the energies $E_a$, $E_d$ and the energy gap of the semiconductor $E_g$ from the final expression of the Hall coeffcient?

Would it also be possible to find the donor and acceptor densities from the Hall coefficient $R_H (T)$ final expression?

Thank you.

Your approach to finding the temperature dependence of the Hall coefficient in semiconductors is generally correct. However, there are a few things to keep in mind:

1. Neglecting temperature dependence of mobilities: This assumption is valid only at low temperatures, where the temperature dependence of mobilities is small. At higher temperatures, the mobilities can vary significantly and should be taken into account in the expression for Hall coefficient.

2. Neglecting acceptor and donor densities: In some cases, the acceptor and donor densities can be neglected, but it is not always the case. These densities can have a significant impact on the overall behavior of the semiconductor and should be considered in the calculations.

3. Calculating energies and energy gap: The energies $E_a$ and $E_d$ can be calculated from the expression for Hall coefficient by making certain assumptions about the donor and acceptor densities. The energy gap $E_g$ can also be calculated from the Hall coefficient, but it is a more complex process and may require additional information about the semiconductor material.

4. Finding donor and acceptor densities: It is possible to calculate the donor and acceptor densities from the Hall coefficient, but it is a challenging task. It requires a detailed understanding of the semiconductor material and the assumptions made in the calculations. It is always better to have direct measurements of these densities rather than relying on the Hall coefficient.

In conclusion, your approach is correct, but it is important to consider the limitations and assumptions involved. It is always better to have direct measurements of the relevant parameters rather than relying on indirect calculations.

## 1. What is the Hall coefficient in semiconductors?

The Hall coefficient is a measure of the strength and direction of the magnetic field induced by an electric current in a semiconductor material. It is defined as the ratio of the induced electric field to the applied magnetic field.

## 2. How does the Hall coefficient change with temperature in semiconductors?

The Hall coefficient in semiconductors is known to change with temperature. As the temperature increases, the Hall coefficient decreases due to the increase in the number of charge carriers and their mobility.

## 3. What is the relationship between the Hall coefficient and the band gap in semiconductors?

The Hall coefficient is directly proportional to the band gap in semiconductors. This means that as the band gap increases, the Hall coefficient also increases.

## 4. What factors can affect the dependence of Hall coefficient on temperature in semiconductors?

The dependence of Hall coefficient on temperature in semiconductors can be affected by various factors such as impurities, defects, and doping levels in the material. These can alter the number and mobility of charge carriers, thus impacting the Hall coefficient.

## 5. How is the Hall coefficient measured in semiconductors?

The Hall coefficient can be measured by applying a magnetic field perpendicular to the direction of current flow in a semiconductor material and measuring the induced electric field. The ratio of these two values gives the Hall coefficient.

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